BALTIC-WAY 2024 Problem 2

Let $\mathbb{R}^{+}$be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that

$$\frac{f(a)}{1+a+c a}+\frac{f(b)}{1+b+a b}+\frac{f(c)}{1+c+b c}=1$$

for all $a, b, c \in \mathbb{R}^{+}$that satisfy $a b c=1$.